Consider a many-body problem, such as one involving interacting spins or particles, for which the available data, whether experimental or numerical, allows us to determine the corresponding averages and two-point correlation functions. Suppose the distribution of the spin values or of the particle positions is written in a Boltzmann form with one- and two-body interactions only. One question we wish to adress is how to choose the corresponding couplings so that the distribution generates the correct two-point correlations. Such inverse problems are of course ubiquitous in the study of complex systems from simple fluids to neurons or proteins (see for instance [1]}, [2]}, [3]}, [4]}, [5]}, [6]}). In general, making analytical progress for inverse problems requires to resort to approximation schemes. Even when only the averages of the degrees of freedom are constrained this is a formidable task equivalent to solving a statistical mechanical model with a one-body external field. Outside the realm of Gaussian theories which are exactly solvable, few generic methods exist [7]}, [8]}, [9]}, [10]}, [11]}. If one further constrains correlations to be fixed, the problem has been adressed at the field-theoretic level (that rests on an expansion around a Gaussian theory) and is known under various names: Luttinger and Ward, or 2PI, or effective potential methods [12]}, [13]}. Within the realm of simple liquids, the Poisson nature of statistics of the ideal gas has also been exploited [14]}, [15]}, [16]} to derive a one-to-one connection between the pair potential and the pair distribution function. For Ising spins (binary variables), weak correlation expansions have been carried out as well [4]}. However, it is our feeling that, however remarkable, these inroads into specific systems, such as simple liquids or Ising spins, could benefit a more general approach, applicable, e.g. to Potts spins or even, more speculatively, to trajectory-dependent observables [18]}, [19]}, [20]}.
